Department of Electrical Engineering
Institutionen för systemteknik
Master’s thesis
Virtual Vehicle Pitch Sensor
Master’s thesis performed in Automatic Control at Linköping University
by
Hamdi Bawaqneh
LiTH-ISY-EX--11/4472--SE Linköping 2011
Department of Electrical Engineering Linköpings tekniska högskola
Linköping University Linköpings universitet
Virtual Vehicle Pitch Sensor
Master’s thesis performed in Automatic Control
at Linköping University
by
Hamdi Bawaqneh
LiTH-ISY-EX--11/4472--SE
Supervisor: Ph. Stud. Tohid Ardeshiri
isy, Linköping University
M.Sc. Eng. Jonas Josefsson
NIRA Dynamics AB
Examinator: Professor Fredrik Gustafsson
isy, Linköping University Linköping, 27 May, 2011
Division, Department
Avdelning, Institution
Division of Automatic Control Department of Electrical Engineering Linköping University
SE-581 83 Linköping, Sweden
Date Datum 2011-05-27 Language Språk Swedish/Svenska English/Engelska Report category Rapporttyp Licentiate thesis Master’s thesis C-paper D-paper Other
URL for the electronic version
http://www.control.isy.liu.se http://www.ep.liu.se ISBN — ISRN LiTH-ISY-EX--11/4472--SE
Title of series, numbering
Serietitel och serienummer
ISSN
—
Title
Titel
Virtual Vehicle Pitch Sensor Virtuell Pitch Sensor
Author
Författare
Hamdi Bawaqneh
Abstract
Sammanfattning
An indirect tire pressure monitoring system uses the wheel rolling radius as an indicator of low tire pressure. When extra load is put in the trunk of a car, the load distribution in the car will change. This will affect the rolling radius which in its turn will be confused with a change in the tire pressure. To avoid this phenomenon, the load distribution has to be estimated. In this thesis methods for estimating the pitch angle of a car and an offset in the pitch angle caused by changed load distribution are presented and when an estimate is derived, a load distribution can be derived. Alot of available signals are used but the most important are the longitudinal accelerometer signal and the acceleration at the wheels derived from the velocity of the car. A few ways to detect or compensate for a non-zero road grade are also presented. Based on the estimated offset, a difference between the front and rear axle heights in the vehicle can be estimated and compensating for the changed load distribution in an indirect tire pressure monitoring system will be possible.
Keywords
Nyckelord Load distribution, Pitch angle offset, Longitudinal accelerometer, Road grade, Axle height
Abstract
An indirect tire pressure monitoring system uses the wheel rolling radius as an indicator of low tire pressure. When extra load is put in the trunk of a car, the load distribution in the car will change. This will affect the rolling radius which in its turn will be confused with a change in the tire pressure. To avoid this phenomenon, the load distribution has to be estimated. In this thesis methods for estimating the pitch angle of a car and an offset in the pitch angle caused by changed load distribution are presented and when an estimate is derived, a load distribution can be derived. Alot of available signals are used but the most important are the longitudinal accelerometer signal and the acceleration at the wheels derived from the velocity of the car. A few ways to detect or compensate for a non-zero road grade are also presented. Based on the estimated offset, a difference between the front and rear axle heights in the vehicle can be estimated and compensating for the changed load distribution in an indirect tire pressure monitoring system will be possible.
Sammanfattning
Ett indirekt däcktrycksövervakningssystem använder sig av existerande sensorer i fordonet för att estimera förändringar i däcktryck. Huvudprinciperna för att upptäcka när ett eller flera däck förlorar lufttryck bygger på däckets förändringar i rullradie. I många fall finns andra yttre omständigheter som påverkar däcket på ett liknande sätt som en tryckförlust och för att förbättra prestandan på indirekta däcktrycksövervakningssystem används olika sensorer i bilen för att kompensera för dessa fall. Exempel på ett fall som påverkar däcket på ett liknande sätt som en tryckförlust är när en stor ilastning görs i bagageutrymmet på en bil. Extra lasten kommer att ge en ändrad massfördelning vilket kommer att få rullradien att minska på samma sätt som en tryckförlust. Om man lyckas skatta massfördelningen så kan denna information med fördel användas för att kompensera för lastförändringar. I detta examensarbete kommer olika metoder för att skatta offseten i pitchvinkeln att presenteras. När en skattning är framtagen kommer det att vara möjligt att räkna ut massfördelningen i bilen när man känner till fjäderkonstanten. Exempel på problem är en nollskild vägvinkel som måste upptäckas eller kompenseras bort.
Acknowledgements
I would like to thank my supervisor at NIRA Dynamics AB, Jonas Josefsson, for being patient while dealing with my (many) questions.
I would also like to thank my supervisor at Linköping University, Tohid Ardeshiri, for his help and ideas throughout my thesis work.
I’m also thankful for the help I got from Henrik Svensson, Andreas Hall and Thomas Gustafsson at NIRA Dynamics AB.
Last but not least I would like to thank the wonderful and very hard-working staff at NIRA Dynamics AB for their support and interest and for making it a pleasent stay for me; it was a pleasure meeting you.
Hamdi Bawaqneh, Linköping 2011.
Contents
1 Introduction 1 1.1 NIRA Dynamics . . . 1 1.2 Objective . . . 1 1.3 Methods . . . 2 1.4 Thesis Outline . . . 4 1.5 Related work . . . 4 1.6 Test data . . . 5 1.7 Notations . . . 72 First approaches of estimating the offset 9 2.1 First approach of pitch angle and pitch angle offset estimation . . 9
2.2 Power Spectral Density Analysis . . . 14
2.3 Offset Estimation Stopping Conditions . . . 16
3 Finding vertical/pitch dynamics 17 3.1 Identifying unknown parameters . . . 18
3.2 Estimating the pitch angle . . . 19
3.3 Getting the offset using a model . . . 20
3.4 Estimating parameters using Grey Box model . . . 22
4 Compensating for a road grade 25 4.1 Pitch angle vs Road grade . . . 25
4.2 Detecting a hill . . . 28
4.2.1 Using the longitudinal accelerometer signal . . . 28
4.2.2 Using the torque signal . . . 29
5 Comparison of the methods 33 6 Conclusions and future work 35 6.1 Conclusions . . . 35
6.2 Future work . . . 36
Bibliography 39
Chapter 1
Introduction
1.1
NIRA Dynamics
This thesis is done at NIRA Dynamics where their main product is an indirect tire pressure monitoring system, TPI.
TPI is a software solution which detects changes in the tire inflation pressure using sensor fusion of the available information in a vehicle where the wheel speed signals are the main input signals. TPI does not require any in-wheel pressure sensors and RF (Radio Frequency) components. TPI is capable of detecting pressure drops in one, two, three, or even up to all four tires. After detection of a low pressure condition, the system can point out (isolate) which tire or tires that are under-inflated.
1.2
Objective
When extra load is put in a car, the load distribution in that car will change, which in its turn will affect the rolling radius of the wheels. The change in the rolling radius in its turn can be confused with a change in the tire inflation pressure. To avoid this phenomenon, the load distribution has to be estimated. The main goal of this thesis is to estimate the pitch angle offset δ seen in Figure 1.1, caused by a changed load distribution. Based on the obtained estimation, compensating for the changed load distribution in TPI will be possible.
The influence of a changed load distribution on the rolling radius is clearly seen in Figure 1.2 where the blue/green lines describe the change for the front wheels and the red/pink lines describe the change for the rear wheels. In the interval 600-800 minutes a heavy load is added to the trunk and the rolling radius of the rear wheels decreases.
Figure 1.1. The pitch angle θ contains a dynamic part θdand a constant offset δ. One
of the external disturbances that affects the longitudinal accelerometer is the road grade
αs.
1.3
Methods
In this thesis some signals are assumed to be known and these signals are listed in Table 1.1.
Known Signals
Velocity
Acceleration at the wheels Longitudinal acceleration Engine torque
Engine RPM Yaw Rate
Lateral acceleration Steering wheel angle Wheel angular velocity
Table 1.1. Known signals.
Signals that are assumed to be unknown when estimating the offset are listed in Table 1.2.
There are a lot of issues that need to be solved to reach the goal of estimating the pitch angle offset, for example a non-zero road grade affects the longitudinal accelerometer in the same way as a longitudinal acceleration that yields a pitch movement and combining several methods may be necessary. Other difficulties that can affect the results are for example the vehicle geometry, suspension settings and propulsion type. If a car has a very long wheel base and very stiff springs, the effect of an added load in the trunk on the longitudinal accelerometer will not be
1.3 Methods 3
(a) Change in vehicle load
(b) Scaled change in rolling radius of the four wheels.
Figure 1.2. Figure (a) shows the change in vehicle load and (b) shows the change in the
rolling radius, where the blue/green lines describe the change for the front wheels and the red/pink lines describes the change for the rear wheels. When adding a heavy load to the trunk the rolling radius of the rear wheels will decrease which can be seen in the interval 600-800 s. The two figures are obviously correlated and by estimating the offset in the pitch angle and calculating the load distribution, compensation for the effect of changed load distribution on the rolling radius can be done, which will make the TPI system more robust.
Unknown Signals
Absolute position Front axle height Rear axle height Vertical acceleration
Table 1.2. Unknown signals.
big and the estimation of the pitch angle offset will become more difficult. The effect of the temperature on the longitudinal accelerometer is another problem since the bias error in the longitudinal accelerometer is temperature dependent. However, the last mentioned problem will not be considered in this thesis. Moreover, the plots in this thesis will have two different x-axis labels, ’Time’ when all data in a data set have been used and ’Samples’ when only chosen data have
been used.
1.4
Thesis Outline
Chapter 2 - First approaches of estimating the offset:
First approaches of estimating the pitch angle and the pitch angle offset are pre-sented. Here, it is assumed that the road grade is zero.
Chapter 3 - Finding vertical/pitch dynamics:
A more advanced method to estimate the pitch angle and the offset is presented. Here, it is also assumed that the road grade is zero.
Chapter 4 - Compensating for a road grade:
Approaches for detecting a non-zero road grade are presented and the possibility of compensating for the road grade is discussed.
Chapter 5 - Comparison of the methods:
A short comparison between the methods that gave satisfactory results is made.
Chapter 6 - Conclusions and future work:
Conclusions and recommendations for future work are presented.
1.5
Related work
The problem of estimating the pitch angle in a car and the road gradient is not a new problem. However there aren’t that many papers dealing with the problem of estimating the pitch angle offset.
There has been some internal work at NIRA Dynamics AB to estimate the pitch angle offset, for example by using the engine torque signal or the longitudinal ac-celeration but the results were not satisfactory.
In this thesis, the offset is derived based on an estimated pitch angle and the methods to achieve this goal are many. However, many of these methods assume that some specific conditions are fullfilled and many others use signals that are considered unavailable in this thesis such as the absolute position and the vertical accelerometer.
In a paper [2] by Tseng, Xu and Hrovat a method that uses an IMU (Inertial Measurement Unit) in estimating the vehicle roll and pitch angles is presented. The method achived accurate estimation of both pitch and roll angles.
One of the measurements that the IMU outputs is the vertical acceleration which means that this method is not suitable for this thesis since the vertical acceleration is considered not available here. However, the article also presents a simple way to make a rough estimate of the vehicle pitch angle which will be described later in this thesis.
Another method is [5] by Ryu and Gerdes where they assume that the vehicle pitch is caused mostly by a road grade and the road grade is estimated based on vertical and horizontal velocity from a GPS.
Also in [3] by Hong the GPS is used and two methods for obtaining an estimate of the road grade are demonstrated.
1.6 Test data 5
the engine torque are combined with observations of the road altitude from a GPS receiver in a Kalman filter to form a road grade estimate based on a system model. However, the torque delivered from the engine suffers from many losses on its way through the transmission down to the ground [6] and an accurate model of it is very difficult to achieve.
A report by Massel, Ding and Arndt [13] presents different approaches to estimate the road uphill gradient and the vehicle pitch angle with the help of the vertical accelerometer, the longitudinal accelerometer, the wheel speed sensors and static mathematical models.
Lundquist and Schön provide a way for identifying the pitch dynamics of a ve-hicle [9]. The method is based on a dynamic longitudinal and vertical motion model of a car and solving a gray-box problem, assuming that the road grade is zero.
There are many patents in this application area as well. Examples of patents are [4] by Steenson and [1] by Hrovat. Steenson [4] uses an apparatus and a method to estimate the pitch state with the help of gyroscopes and at least one inertial reference sensor. The patent by Hrovat [1] is especially interesting since it provides a method for estimating the road grade/vehicle pitch by using commonly available signals in vehicles such as the steering wheel angle sensor, yaw rate sensor, wheel speed sensors, lateral acceleration sensor, longitudinal acceleration sensor and roll rate sensor. However, in Hrovats patent the pitch angle of the vehicle corresponds to the grade of the road when the vehicle wheels are on the road without significant suspension motion.
1.6
Test data
In this section I’ve chosen to summarize the used data from different driving tests in this thesis. Table 1.3 shows the loads used when driving in Linköping-Sweden several routes with an Audi A7. Table 1.4 shows the loads used when driving at the Neustadt track-Germany with an Audi A7. Table 1.5 shows the loads used when driving in Linköping-Sweden several routes with an Audi A3. Table 1.6 shows the loads used when driving in Death Valley-USA with a VW Passat.
Tests Description of the load change Test 1 0 kg, with a half full fuel tank
Test 2 0 kg, full tank
Test 3 50 kg in trunk, full tank Test 4 100 kg in trunk, full tank Test 5 150 kg in trunk, full tank
Test 6 325 kg = 150 kg in trunk + 175 in back seat, full tank
Test 7 385 kg = 150 kg in trunk + 140 in back seat + 95 in front seat, full tank
Tests Description of the load change Test 1 100 kg in trunk, full tank Test 2 300 kg in trunk, full tank
Table 1.4. The tests above were done in Neustadt-Germany with an Audi A7.
Tests Description of the load change Test 1 0 kg, with a half full fuel tank
Test 2 0 kg, full tank
Test 3 80 kg in front seat, full tank Test 4 80 kg in front seat, full tank Test 5 80 kg in front seat, full tank
Test 6 150 kg = 100 kg in trunk + 50 kg in trunk, full tank Test 7 150 kg = 100 kg in trunk + 50 kg in trunk, full tank Test 8 230 kg = 150 kg in trunk + 80 kg in front seat, full tank Test 9 230 kg = 150 kg in trunk + 80 kg in front seat, full tank
Table 1.5. These tests were done in Linköping-Sweden with an Audi A3.
Tests Description of the load change Test 1 50 kg in back seat, full tank
1.7 Notations 7
1.7
Notations
δ Offset in the pitch angle [rad]
θ Pitch angle [rad]
˙
θ Pitch angular velocity [rad/s]
θd Dynamic part of the pitch angle [rad] ax Longitudinal accelerometer signal [m/s2]
v Longitudinal velocity [m/s]
˙v Longitudinal acceleration, using the wheel speeds [m/s2]
g Gravitational constant [m/s2]
αs Road grade [rad]
Af Vehicle frontal cross area [m2]
cw Drag coefficient [−]
ρ Air density [kg/m3]
Cs Spring constant [N/m]
Cd Damper constant [N s/m]
lr Distance from the CoG to the rear axle (unloaded) [m] lf Distance from the CoG to the front axle (unloaded) [m] I0 Moment of inertia around the y-axis (unloaded car) [kgm2]
M Car mass when unloaded [kg]
m Mass of added load [kg]
h Height of center of gravity [m]
hair Height of center of drag [m]
∆zf vertical position of the front suspension [m]
∆zr vertical position of the rear suspension [m] z The vertical position of the complete chassis [m]
˙
z The vertical velocity of the complete chassis [m/s]
ωc Cut-off frequency [rad/s]
F Tractive force [N ]
Fair Air resistance [N ]
Frolling Rolling resistance [N ]
Fg Longitudinal force of gravity [N ]
Te Engine torque [N m]
Je Engine inertia [N ms2/rad]
Jdl Wheel and propellershaft inertia [N ms2/rad]
ωe Engine angular velocity [rad/s]
ωw Wheel angular velocity [rad/s]
RP M Engine RPM [Revol./min]
it Transfer ratio [−]
r Wheel radius [m]
lm Distance from the extra load to the rear axle [m] lr,new Distance from the new CoG to the rear axle [m]
Chapter 2
First approaches of
estimating the offset
There are two approaches presented in this chapter, an approach with a simple equation and an attempt to use the power spectral density of the estimated pitch angle to reveal a change in the mass has also been done. Here, it is also assumed that the road grade is zero.
2.1
First approach of pitch angle and pitch angle
offset estimation
Figure 2.1. Longitudinal acceleration ax, the acceleration at the wheels ˙v and the pitch
angle θ.
A first approach is done by using Equation 2.2 which is taken from Xu, Tseng and Hrovats paper [2] where they present a simple way to make a rough estimate
of the vehicle pitch angle based on Figure 2.1 as in Equations 2.1 and 2.2: ax= ˙v + gsin(θ) (2.1) θ = arcsin ax− ˙v g ≈ (ax− ˙v) g (2.2)
where θ is the estimated pitch angle, v the longitudinal velocity, ˙v its time derivative and axthe longitudinal accelerometer signal. The approximation above
is valid assuming that the pitch angle is small.
In case of small wheel slip, the vehicle velocity can be derived using the speeds of the four wheels. The acceleration ˙v can then be calculated by differentiating the velocity according to the five point method for the first derivative:
˙v = −v(t − 2T ) + 8v(t − T ) − 8v(t + T ) + v(t + 2T )
12T (2.3)
It’s worth noticing that this differentiating method is non-causal and will delay the signal two samples.
Figure 2.2. The acceleration at the wheels derived by using Equation 2.3 is almost
consistent with the signal from the longitudinal accelerometer.
Figure 2.2 shows the acceleration at the wheels derived by using Equation 2.3 together with the longitudinal accelerometer signal when the load in the car is zero and the slip is small, and it is seen that the two lines are almost consistent. However, the acceleration at the wheels used in this thesis is the one available on the vehicle’s CAN-bus.
2.1 First approach of pitch angle and pitch angle offset estimation 11
The estimate of the vehicle acceleration using wheel speeds is vulnerable since it will be affected by disturbances such as slip or if the pressure in a wheel pair changes. If the car is rear or front wheel driven, only the non-propulsive wheel pair are used in the calculations to avoid the slip. Problems appear when the car is all wheel driven since the effect of the slip will be almost the same on all four wheels. However, one can still compare the left wheel pair with the right wheel pair.
As a first approach of pitch angle estimation, Equation 2.2 will be used when driving straight forward and assuming that the road grade is zero. This is then validated using measured data extracted from a database.
Figure 2.3 shows the estimated pitch angle θ together with the measured pitch angle using axle height sensors.
The measured pitch angle is calculated according to Equation 2.4 where lb is the
distance between the two wheel bases and ∆zf,r are the vertical position of the
front/rear suspension.
θmeasured=
∆zf− ∆zr lb
(2.4)
Figure 2.3. A rough estimate of the pitch angle θ using Equation 2.2 plotted against
the calculated pitch angle using measured data according to Equation 2.4 when the car is unloaded. The estimated pitch angle clearly captures the pitching dynamics, especially for small angles.
By comparing the estimated pitch angle and the measured pitch angle, it is clear that the estimated one captures the pitching dynamics, especially for small angles but not that well when the pitch angle is big.
When the mass of a car is changed, for example some load is added to the trunk, a constant offset in the pitch angle will be introduced due to changed load distribu-tion. As seen in Figure 2.4, both the measured and estimated pitch angles clearly have a constant offset when LP-filtering the signals.
Figure 2.4. An offset in the pitch angle is introduced when load is added to the trunk.
This is clearly seen in both the measured and estimated pitch angles using Equation 2.2 after LP-filtering. The data used here are test2 and test5 in Table 1.3.
One way of estimating the offset is by calculating the mean value of the esti-mated pitch angle when driving straight forward with a non-zero velocity. Figure 2.5 shows how an added load affects the offset and it’s clearly seen that the offset will change when the load is changed, i.e. the load distribution is changed. The data used here were collected while driving an Audi A7 several routes with differ-ent loads.
This simple method where the mean value is calculated has also been validated on a test database where data with different loads placed in different ways in the car had been chosen and the results were promising. However, the results where the car had been driven for a very short time, for example two minutes, were often wrong or misleading which might depend on the fact that the longer time the car is driven, the better accuracy of the mean value is achieved.
Some results were less accurate when the road surface was slippery or when there was gravel on the road since the slip is much higher then, which affects the accel-eration derived using the wheel speeds.
Other cases where it was hard to see a change in the offset were when specific car types were used which might depend on the vehicle geometry and suspension settings.
2.1 First approach of pitch angle and pitch angle offset estimation 13
Figure 2.5. The figure shows the estimated offset using Equation 2.2 when driving with
an Audi A7 in Linköping-Sweden several routes with different loads plotted together with the offset derived using measured data and Equation 2.4. The roads in this specific region can be considered flat and only data samples when driving straight forward have been used. The two lines are almost consistent and the relation between added load and offset is clearly seen when looking at the tests listed in Table 1.3 which were used here.
An alternative way of estimating the offset is to only use data when driving straight forward with a constant non-zero velocity on a flat road, i.e. when ˙v is zero, since we in such a case don’t have a dynamic movement and the offset will be revealed. A disadvantage with the latter method is that there will be less data to work with since we must wait for data samples where the car has constant velocity. Figure 2.6 shows that the pitch angle offset will be revealed when driving straight forward with a constant velocity on a flat road with different loads in the trunk. Worth mentioning here is that air resistance will affect the offset angle estimation, especially for high speeds, but this is not considered in this thesis.
Figure 2.6. The figure shows the estimated offset when driving on a flat road with
constant velocity and different loads in the trunk. The car used here is an Audi A7 that was driven at the Neustadt track in Germany and the tests are listed in Table 1.4.
2.2
Power Spectral Density Analysis
The power spectral density of the estimated pitch angle shown in Figure 2.7(a) was intended to be used for finding frequencies where the order between the lines is clearly seen. For example a car with a heavy load should have more power in the low frequency component and little power in the high frequency components under normal conditions.
However, it’s very difficult to find such distinct areas when observing Figure 2.7(a) which might depend on the fact that different driving styles affects the power spectral density in different ways, for example a dynamic driving style with heavy acceleration and braking will increase the power in the higher frequency compo-nents.
When plotting the PSD for many different test data-sets, a conclusion is taken that only very heavy loads will most of the time be seen when added to the trunk as seen in Figure 2.8.
2.2 Power Spectral Density Analysis 15
(a) Power spectral density
(b) Power spectral density enlarged
Figure 2.7. Power spectral density. When looking at the DC component in Figure
2.7(b) it can be noticed that the curves appear almost in the same order as the load in the car. A low resolution is chosen on purpose for a better illustration. Figure 2.7(a) was intended to find frequencies where the order between the lines is well defined but this is obviously difficult. The car used here is an Audi A3 and the tests are listed in Table 1.5.
Another conclusion is that when a load is added to the front seat, it will be confused with a heavy load added to the trunk when plotting the PSD, especially
Figure 2.8. The power spectral density of the estimated pitch angle. A heavy load in
the trunk has low power in almost all higher frequencies. The car used here is an Audi A7 and the tests are test1-test6 in Table 1.3. The DC part of the signals is shown in Figure 2.5.
for high frequencies.
Since the pitch angle offset is the DC part of the estimated pitch angle, one can simply low-pass-filter the signal to reveal the offset. This can also be seen in Figure 2.7(b) where the amplitudes of the curves in the low frequency component appear almost in the same order as their load.
2.3
Offset Estimation Stopping Conditions
In this section a summary of the pitch angle offset estimation stopping/pausing conditions is presented.
The estimation is stopped/paused whenever one of the following conditions are valid: While turning, when the slip is too big and when there is a non-zero road grade present. Methods for detecting a non-zero road grade will be discussed later in Chapter 4.
Chapter 3
Finding vertical/pitch
dynamics
In this chapter a method used by Lundquist [9] to estimate the pitch angle is presented. Figures 3.1 and 3.2 show the variables used to describe the vertical motion of the vehicle and also the longitudinal forces acting on the vehicle. First, unknown parameters are identified and the model is then validated by comparing the estimated pitch angle with the calculated pitch angle using measured data. The same model is then used to estimate the pitch angle offset.
Figure 3.1. Definition of the variables. The figure is taken from [9] and modified.
Figure 3.2. Vertical and longitudinal forces. The figure is taken from [9].
3.1
Identifying unknown parameters
The method is used to identify unknown parameters such as the spring constant
Cs, the damper constant Cd, the height of center of drag hair, and the distance
from the center of gravity to the front wheel axis lf. The parameters are identified
by using the System Identification Toolbox [8].
Equation (3.1) leading to the state-space model is shown below and the ap-proximation litan(θ) ≈ liθ, i = r, f has been used assuming that the pitch angle
is small. Fzf = −Cs(z − lfθ) − Cd( ˙z − lfθ)˙ (3.1a) Fzr= −Cs(z + lrθ) − Cd( ˙z + lrθ)˙ (3.1b) M ¨z = Fzr+ Fzf (3.1c) M ¨x = Fxf+ Fxr− Fair (3.1d) ¨
θI0= −Fzflf+ Fzrlr− (Fxf+ Fxr)h − Fair(hair− h) (3.1e)
The obtained state-space model describing the system can then be written as in Equation (3.2) where I0 is the moment of inertia around the y-axis when
considering the car to be a solid cuboid [17], cw is the drag coefficient, Af the
car’s frontal cross area, ρ the air density and ∆zf,r is the height differences in the
front/rear axles. ˙ x = 0 1 0 0 −2Cs M − 2Cd M Cslf−Cslr M Cdlf−Cdlr M 0 0 0 1 Cslf−Cslr I0 Cdlf−Cdlr I0 − Csl2f+Csl2r I0 − Cdlf2+Cdl2r I0 x+ 0 0 0 0 0 0 −M h I0 − cwAfρ2hair I0 u (3.2a)
3.2 Estimating the pitch angle 19 y =1 0 −lf 0 1 0 lr 0 x (3.2b) x = z z˙ θ θ˙T (3.2c) y = ∆zf ∆zr T (3.2d) u = ax v2 T (3.2e)
The values of the identified parameters of an Audi A7 is:
Cs= 81600, Cd = 5015, lf = 1.260 and hair = 0.46.
The only available real value in this thesis is lf = 1.227 which means that the
identified lf is good enough. However, a validation is done in Section 3.2 to make
sure that the model with the identified paramters are good enough.
3.2
Estimating the pitch angle
The previously identified parameters can then be used in any dynamic longitudinal and vertical motion model of the car that contains these parameters. A discrete static Kalman filter [7] is then used (Equation (3.3)) together with the state-space model (3.2) to estimate the states when using a validation data sequence. Time update equations:
ˆ
xk= Aˆxk−1+ Buk−1 (3.3a)
Pk = APk−1AT + Q (3.3b)
Measurement update equations:
Kk = PkCT(CPkCT+ R)−1 (3.3c)
ˆ
xk = ˆxk+ Kk(yk− C ˆxk) (3.3d)
Pk = Pk− KkCPk (3.3e)
Here P is the covariance matrix of the estimation error, A is the state transition model, B is the input model, C is the observation model, K is the Kalman gain and Q and R are the noise covariances.
A validation of the model is made by comparing the estimated pitch angle to the raw one as seen in Figure 3.3 and a conclusion is therefore made that the model is good enough, especially when driving straight forward and the lateral acceleration is fairly small.
Figure 3.3. The estimated pitch angle using a model follows the measured one quite
well, especially when a large excitation is present, i.e. when the acceleration and pitch angle are big. The data used here is test2 in Table 1.3.
3.3
Getting the offset using a model
To get the offset for every test data-set using the model described in Section 3.1, one can first use the input signal u = ˙v v2T
to get an estimated pitch angle not containing an offset (since ˙v does not contain any offset) and then use the input signal u = ax v2
T
to get an estimated pitch angle that will be affected by the offset in the longitudinal accelerometer signal ax. When the difference between
these two estimations is calculated and a mean value of that difference is derived, the result will apparently be load dependent as seen in Figure 3.5.
Figure 3.4 shows the principle of this method, where two kalman filters work in parallell and two estimated pitch angles are derived. The mean value of the difference between the estimated pitch angles yields the estimated offset.
When comparing the estimated offset angles using a model with the offset angles derived using measured data one can see that they are almost consistent.
3.3 Getting the offset using a model 21
Figure 3.4. The principle of the method presented in Section 3.3. Two kalman filters
work in parallel and two estimated pitch angles are derived. The mean value of the difference between the estimated pitch angles yields an estimated offset.
Figure 3.5. The estimated offset in the pitch angle using a model is clearly load
depen-dent and almost consistent with the offset using measured data. The data used here are the same tests as in Figure 2.5 and Table 1.3.
3.4
Estimating parameters using Grey Box model
When a load is added to the vehicle mass, the placement of the center of gravity might be affected. In this section the previously presented model will be used to estimate the distance from the rear axle to the new center of gravity.
The estimated parameters, i.e. the spring constant Cs, damper constant Cd, height
of center of drag hair and the distance from the center of gravity to the front wheel
axle when the car is unloaded, lf, will be known constants in this chapter. The
mass of the car will also be a known constant.
After alot of tuning of the data selection conditions and tests perfomed on several data-sets a conclusion is made that this method will not be satisfactory most of the time. This can be explained by the fact that since we can’t guarantee that there will be enough excitation in the acceleration when someone is driving the car, this method will give identified parameters with very different accuracy in different data-sets.
Figure 3.6 shows the identified and calculated distance from the rear wheel axle to the center of gravity when the load is changed. The calculation of the placement of the new center of gravity is done by using Equation 3.4, where m is the added mass and lm is where the extra load is put in the car. Here we assume that the
load is added to the trunk.
The estimated distance has a tendency of following the calculated distance. How-ever, a good initial guess for lr,new when using a Grey Box model is crucial to get
a good result. The tests listed in Table 1.3 are used here.
lr,new =
M lr+ mlm
3.4 Estimating parameters using Grey Box model 23
Figure 3.6. Distance from the rear wheel axle to the center of gravity when the load
is changed, identified using Grey Box Model and calculated using Equation 3.4. The estimated distance has a tendency of following the calculated distance. The data used here are the tests in Table 1.3.
Chapter 4
Compensating for a road
grade
The effect of a non-zero road grade has not been discussed in this thesis so far. In this chapter, methods for detecting a non-zero road grade is presented and the possibility of compensating for a non-zero road grade is discussed as well. The rough estimate of the pitch angle in Equation 2.2 was also used in [14] to estimate the road grade assuming that the pitch angle is small enough and the road grade could be determined with a resolution of under 2.25 degrees.
4.1
Pitch angle vs Road grade
To calculate the road grade, measured elevation data extracted from the Global Digital Elevation Model (GDEM) provided by NASA and METI was used. The resolution is generally between 15 and 90 meters [10] but the resolution of the USA data is higher and is approximately between 3 and 30 meters [16].
The road grade is calculated when the car has a non-zero velocity and driving straight forward simply by dividing the difference in elevation with the change in driven distance.
Elevation data from a road trip in Death Valley in USA is shown in Figure 4.1 and the corresponding road grade data is shown in Figure 4.2.
Figure 4.1. Elevation data from a test done in the Death Valley (USA), i.e. the test in
Table 1.6.
Figure 4.2. Corresponding road grade data calculated from the data in Figure 4.1 when
driving straight forward with a non-zero velocity after LP-filtering. Only the sum of the road grade and pitch angle can be calculated from the accelerometer data and the data derived from the accelerometer has almost the same shape as the road grade derived from elevation data.
In Josefssons work [6] the road grade was modeled as a first order model with a cut-off frequency ωc. The problem of separating the road grade αsfrom the offset
in the pitch angle δ caused by an added load can be described in the state-space model 4.1: ˙ x =−ωc 0 0 0 x + ν (4.1a)
4.1 Pitch angle vs Road grade 27
x = αs δ
T
(4.1b)
y = αs+ δ = 1 1 x (4.1c)
Where y is obtained by only using data when driving straight forward with constant velocity. In Schmidtbauer and Lingmans work [11] a spatial cut-off fre-quency wc = 2π · 0.002 rad/s was chosen for representing the road slope and the
same cut-off frequency is used in Figure 4.3 where the result of using the model 4.1 with the Kalman filter 3.3 is shown.
Figure 4.3. Estimated road grade using the state-space model 4.1 vs calculated road
grade using elevation data in the upper plot and the estimated offset in the lower one. Here, only data when driving straight forward with constant velocity has been used. The data used here is the same as in Table 1.6.
However, the most suitable cut-off frequency depends on what the road profile looks like. For example a road in Death Valley as in Figure 4.1 with many long and steep hills would have a different cut-off frequency than a road with many short hills (also in Death Valley) as shown in the marked area in Figure 4.4.
Since we don’t know what the road profile looks like, we cannot determine the cut-off frequency wc and the model would give different accuracy on different road
profiles. A solution to this problem is to stop the estimation of the offset or to estimate the road grade whenever a change in the road grade is detected and the ways of compensating for, or detecting, a hill will be discussed in more detail in section 4.2.
Figure 4.4. A road profile with many short hills will require a different cut-off frequency
than a road with many long and steep hills. The Elevation data shown here is the same as in Figure 4.1 where an area with many small hills is marked with a rectangle.
4.2
Detecting a hill
In this section a couple of methods of detecting a non-zero road grade are presented.
4.2.1
Using the longitudinal accelerometer signal
The maximum and minimum offset in the pitch angle due to changed load in the car can be derived by simply standing still on a flat road surface and putting the maximum allowed load in the trunk and in the front seats respectively. When driving with constant speed the expression ax
g = αs+ δ is valid since the dynamic
part of the pitch angle will be equal to zero, see Figure 1.1.
Whenever the value of this expression is not within the limits, i.e. less than the minimum offset value or larger than the maximum offset value, the estimation of the offset is stopped.
A disadvantage with this simple method is that the estimation of the offset will not be stopped if the road grade is small, only if it is big enough to give a value of αs+ δ that is outside the limits. Another disadvantage is that the effects of air
resistance and constant acceleration is not included.
Figure 4.5 shows that this method is good enough for detecting hills with a big road grade.
The effect of different temperatures on the bias error in the longitudinal ac-celerometer is not taken in consideration here.
4.2 Detecting a hill 29
Figure 4.5. Whenever a big hill is detected, the pitch angle offset estimation will be
stopped. The data in the figure above were collected while driving straight forward with a non-zero velocity during the test mentioned in Table 1.6 and it can be seen that it is possible to detect a big enough hill, i.e. a hill where the road grade is big enough.
Moreover, this method can theoretically be combined with the method described in Section 4.1 but this is not investigated further.
4.2.2
Using the torque signal
In this section the torque signal is used to estimate the road grade.
The longitudinal dynamics of a car when not braking can be described with Equa-tions 4.2, 4.3, 4.4, 4.5 and 4.6: M ˙v = Fe− Fair− Frolling− Fg (4.2) Fe= (Te− Jeω˙e)it− Jdlω˙w r (4.3) Fair= cwAf ρ 2v 2 (4.4) Frolling = CrM g cos (αs) ≈ CrM g (4.5) Fg= M g sin (αs) ≈ M gαs (4.6)
where Feis the effective engine force at the wheels, Fairthe air resistance force, Frollingthe rolling resistance, Fgis the component of the force of gravity, Teis the
engine torque, Je the engine inertia, ˙ωe the engine angular acceleration, ˙ωw the
wheel angular acceleration, r the wheel radius, Jdlis a constant, it=RP Mωw , cwis
the drag coefficient, Af the car’s frontal cross area and ρ the air density.
In Josefssons thesis [6] the expression Cr= 0.0136 + 0.410−7v2 is used for speeds
below 150 km/h and the same expression is used here without any deeper inves-tigation.
When driving with constant velocity, Equation 4.3 can be simplified and written as in Equation 4.7:
Fe= Teit
r (4.7)
To calculate the effective engine force at the wheels the nominal wheel radius is used. However, the nominal wheel radius is only approximately known and it varies with acceleration, tire pressure etc.
Delays in the power train are also hard to model, internal work at NIRA Dynamics AB gave the conclusion that the delay between torque and accelerometer reading is about 0-0.5 seconds.
The calculations lead to Equation 4.8 where an expression for calculating the road grade is derived.
αs=
Teirt − Afcwρ2v2
M g − Cr (4.8)
By using Equation 4.8 to calculate the road angle αs and combining it with
the expression ax
g = αs+ δ when driving at constant speed, an expression for
calculating the offset δ in the pitch angle due to changed load distribution, can be derived as in Equation 4.9.
In this thesis Equation 4.8 where the road grade is estimated is used to detect a big road grade and in such case pause the estimation of the pitch angle offset.
δ = ax
g + Cr−
Teirt − Afcwρ2v2
M g (4.9)
Figure 4.6 shows the calculated road grade using measured elevation data and the estimated road grade derived by using Equation 4.8 when driving with constant velocity, using the torque signal where the mean value of the torque signal is substracted. By comparing the two lines it is clear that the estimation is good enough. However, problems appear when the torque signal has a negative value because it is very difficult to achieve an accurate model of the negative torque, but this will not be considered in this thesis.
Other disturbances are for example air resistance, braking, different road friction coefficients on different road surfaces and slip that might affect the estimation of the velocity.
The estimated offset angles when using Equation 4.9 and the same data-set as in Table 1.3 are shown in Figure 4.7. The estimated offsets are clearly load dependent and this method gives a result that is similar to the offset angles using measured data.
4.2 Detecting a hill 31
Figure 4.6. Estimated road grade using the torque signal, i.e. Equation 4.8, vs measured
road grade derived from elevation data when driving straight forward with constant velocity. One can see that the missmatch is greater for negative values, i.e. when the torque signal is negative. The test in Table 1.6 was used here.
Figure 4.7. Estimated offset when using the torque signal in Equation 4.9 and the same
data-set as in Table 1.3. The estimated offset angles are clearly load dependent and the result is considered satisfactory when comparing it to the offset angles derived from measured data.
Chapter 5
Comparison of the methods
In this chapter a short comparison between the approaches that gave satisfactory results is done. Three methods gave estimated offset angles that were satisfactory, one simple method presented in Section 2.1 , another more advanced one where a model is used presented in Section 3.3 and a third method where the torque is used presented in Section 4.2.2. The results from these three methods are plotted in Figure 5.1 together with the offset angles obtained using measured data.
When comparing the estimated offset angles using the simple method and the method using a model respectively, it can be seen that they gave almost the same results and that the results are very close to the measured offset angles.
The method where the torque is used gave good results as well but differed more from the measured offset angles.
Figure 5.1. Estimated offset angles from three different methods plotted together with
the offset angles derived using measured data. The tests used here are listed in Table 1.3. The estimated offset angles are consistent with the measured offset angles.
Chapter 6
Conclusions and future work
6.1
Conclusions
One simple solution for estimating the pitch angle offset has been presented. The solution is based on a simple equation and gave good results when validated on a test database. A second solution using a model of the vehicle pitching dynamics has also been presented. This method is more advanced than the first one and gave good results as well but will require more memory capacity when implemented. An attempt of using the power spectral density of the estimated pitch angle was also done. The power spectral density was intended to be used for finding fre-quencies where the order between the lines is clearly seen and has almost the right order but a conclusion is taken that only very heavy loads will be clearly seen when added to the trunk.
Another conclusion is that when a heavy load is added to the front seat, it will be confused with a heavy load added to the trunk when plotting the power spectral density of the estimated pitch angle, i.e this method cannot separate a heavy load added to the front seat from a heavy load added to the trunk.
There has also been an attempt of using a Grey Box model to estimate the place-ment of the new center of gravity when extra load is added but this did not give results that were satisfactory.
These methods are all vulnerable to disturbances in the acceleration at the wheels such as slip and also vulnerable to air resistance since the effect of this force on the offset estimation when driving with high velocity is not known.
Other problems with the pitch angle offset estimation appear when there is a non-zero road grade present and to avoid or reduce the effects of a road grade, some conditions to stop the offset angle estimation had to be used. One solution was to use the velocity together with the torque signal when driving with constant velocity to estimate the road grade and then use the derived road grade estimation to stop the offset estimation whenever a big enough road grade is detected. The road grade estimation gave quite good results but some problems with this method are for example the fact that it is very difficult to achieve an accurate model of
the negative torque or the air resistance when the wind speed is non-zero. Air re-sistance and roll rere-sistance models are inexact and sensitive to external variations. Since an added load affects the pitch angle offset, this will change the effect of the air resistance on the car.
The torque signal was also used to compensate for a non-zero road grade and then estimate the offset and the result is consistent with the estimations when using the simple method and the method where a model is used.
To calculate the effective engine force at the wheels the nominal wheel radius is used. However, the nominal wheel radius is only approximately known and it varies with acceleration, tire pressure etc. Delays in the power train or delays between torque and accelerometer reading are also difficult to model.
A simple method of detecting a non-zero road grade is to use the longitudi-nal accelerometer to reject any estimated value of the offset that is outside some known limits of the offset value and in such case assume that this is caused by a big enough road grade.
This simple method gave quite good results when driving with constant speed. A disadvantage with this simple method is that the estimation of the offset will not be stopped if the road grade is small, only if it is big enough to give a value of the offset that is outside some known limits.
Another method is to use a model with a specific cut-off frequency to estimate the road grade but this will not give accurate results since the road profile is not known and the most suitable cut-off frequency can’t be determined.
The effect of the temperature on the longitudinal accelerometer, i.e. the changing bias error in this sensor due to changed temperature, is not considered in this thesis.
Suspension and vehicle configuration affects the pitch angle sensitivity to load changes and might make it difficult to see a change in the pitch angle offset.
6.2
Future work
Sensors are becoming cheaper and the implementation of an IMU in modern cars will be possible in the future. This will give big opportunities for estimating the pitch angle offset. The vertical accelerometer in the IMU can be used to esti-mate the road grade which will give more data samples to work with instead of stopping the estimation of the offset when a big road grade is detected and will also make it easier and faster to estimate the pitch angle offset. The IMU does also contain a gyro that can be used to calculate the pitch angle and also the offset. An estimation of the absolute mass together with an estimation of the offset will make it possible to calculate the exact mass distribution in the car. For the abso-lute mass estimation one can for instance model the braking force on the wheels when the clutch pedal is held down with an ARMA model [15], or use a longitudi-nal dynamic model when the car is accelerating to estimate the absolute mass [6].
6.2 Future work 37
The effect of the temperature on the longitudinal accelerometer is not considered in this thesis but a solution might be to use an approximative function with the temperature as input for estimating the bias error in the longitudinal accelerome-ter.
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